Difference between revisions of "TaylorPolynomial Command"

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<noinclude>{{Manual Page|version=5.0}}</noinclude>{{command|cas=true|function}}
{{command|function}}
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; TaylorPolynomial( <Function>, <x-Value>, <Order Number> )
; TaylorPolynomial[ <Function>, <Number a>, <Number n>]
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:Creates the power series expansion for the given function at the point ''x-Value'' to the given order.
:Creates the power series expansion for the given function about the point ''x = a'' to order ''n''.
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:{{example| 1=<code><nowiki>TaylorPolynomial(x^2, 3, 1)</nowiki></code> gives ''9 + 6 (x - 3)'', the power series expansion of ''x<sup>2</sup>'' at ''x = 3'' to order ''1''.}}
:{{example| 1=<div><code><nowiki>TaylorPolynomial[x^2, 3, 1]</nowiki></code> gives ''6 x - 9'', the power series expansion of ''x<sup>2</sup>'' at ''x = 3'' to order ''1''.</div>}}
 
 
==CAS Syntax==
 
==CAS Syntax==
; TaylorPolynomial[ <Function>, <Number a>, <Number n>]
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; TaylorPolynomial( <Expression>, <x-Value>, <Order Number> )
:Creates the power series expansion for the given function about the point ''x = a'' to order ''n''.
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:Creates the power series expansion for the given expression at the point ''x-Value'' to the given order.
:{{example| 1=<div><code><nowiki>TaylorPolynomial[x^2, a, 1]</nowiki></code> gives ''-a<sup>2</sup> + 2 a x'', the power series expansion of ''x<sup>2</sup>'' at ''x = a'' to order ''1''.</div>}}
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:{{example| 1=<code><nowiki>TaylorPolynomial(x^2, a, 1)</nowiki></code> gives ''a<sup>2</sup> + 2a (x - a)'', the power series expansion of ''x<sup>2</sup>'' at ''x = a'' to order ''1''.}}
;TaylorPolynomial[ <Function>, <Variable>, <Number a>, <Number n>]
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;TaylorPolynomial( <Expression>, <Variable>, <Variable Value>, <Order Number> )
:Creates the power series expansion for the given function with respect to the given variable about the point ''Variable = a'' to order ''n''.
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:Creates the power series expansion for the given expression with respect to the given variable at the point ''Variable Value'' to the given order.
:{{example| 1=<div><code><nowiki>TaylorPolynomial[x^3 sin(y), x, 3, 2]</nowiki></code> gives ''sin(y) (9 x<sup>2</sup> - 27 x + 27)'', the power series expansion with respect to ''x'' of ''x<sup>3</sup> sin(y)'' at ''x = 3'' to order ''2''.</div>}}
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:{{examples| 1=<div>
:{{example| 1=<div><code><nowiki>TaylorPolynomial[x^3 sin(y), y, 3, 2]</nowiki></code> gives ''<math>\frac{cos(3) x^{3} (2 y - 6) + sin(3) x^{3} (-y^{2} + 6 y - 7)}{2}</math>'' , the power series expansion with respect to ''y'' of  ''x<sup>3</sup> sin(y)'' at ''y = 3'' to order ''2''.</div>}}
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:*<code><nowiki>TaylorPolynomial(x^3 sin(y), x, 3, 2)</nowiki></code> gives ''27 sin(y) + 27 sin(y) (x - 3) + 9 sin(y) (x - 3)<sup>2</sup>'', the power series expansion with respect to ''x'' of ''x<sup>3</sup> sin(y)'' at ''x = 3'' to order ''2''.
{{note| 1=The order n has got to be an integer greater or equal to zero.}}
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:*<code><nowiki>TaylorPolynomial(x^3 sin(y), y, 3, 2)</nowiki></code> gives ''x<sup>3</sup> sin(3) + x<sup>3</sup> cos(3(y - 3) - x<sup>3</sup> <math>\frac{sin(3) }{2}</math> (y - 3)<sup>2</sup>'', the power series expansion with respect to ''y'' of  ''x<sup>3</sup> sin(y)'' at ''y = 3'' to order ''2''.</div>}}
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{{note| 1=The order has got to be an integer greater or equal to zero.}}

Latest revision as of 08:55, 9 October 2017


TaylorPolynomial( <Function>, <x-Value>, <Order Number> )
Creates the power series expansion for the given function at the point x-Value to the given order.
Example: TaylorPolynomial(x^2, 3, 1) gives 9 + 6 (x - 3), the power series expansion of x2 at x = 3 to order 1.

CAS Syntax

TaylorPolynomial( <Expression>, <x-Value>, <Order Number> )
Creates the power series expansion for the given expression at the point x-Value to the given order.
Example: TaylorPolynomial(x^2, a, 1) gives a2 + 2a (x - a), the power series expansion of x2 at x = a to order 1.
TaylorPolynomial( <Expression>, <Variable>, <Variable Value>, <Order Number> )
Creates the power series expansion for the given expression with respect to the given variable at the point Variable Value to the given order.
Examples:
  • TaylorPolynomial(x^3 sin(y), x, 3, 2) gives 27 sin(y) + 27 sin(y) (x - 3) + 9 sin(y) (x - 3)2, the power series expansion with respect to x of x3 sin(y) at x = 3 to order 2.
  • TaylorPolynomial(x^3 sin(y), y, 3, 2) gives x3 sin(3) + x3 cos(3) (y - 3) - x3 \frac{sin(3) }{2} (y - 3)2, the power series expansion with respect to y of x3 sin(y) at y = 3 to order 2.
Note: The order has got to be an integer greater or equal to zero.
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